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2021 Rigidity in étale motivic stable homotopy theory
Tom Bachmann
Algebr. Geom. Topol. 21(1): 173-209 (2021). DOI: 10.2140/agt.2021.21.173


For a scheme X, denote by 𝒮(Xét) the stabilization of the hypercompletion of its étale –topos, and by 𝒮ét(X) the localization of the stable motivic homotopy category 𝒮(X) at the (desuspensions of) étale hypercovers. For a stable –category 𝒞, write 𝒞p for the p–completion of 𝒞.

We prove that under suitable finiteness hypotheses, and assuming that p is invertible on X, the canonical functor

ep:𝒮(X ét)p𝒮 ét(X)p

is an equivalence of –categories. This generalizes the rigidity theorems of Suslin and Voevodsky (Invent. Math. 123 (1996) 61–94), Ayoub (Ann. Sci. École Norm. Sup. 47 (2014) 1–145) and Cisinski and Déglise (Compos. Math. 152 (2016) 556–666) to the setting of spectra. We deduce that under further regularity hypotheses on X, if S is the set of primes not invertible on X, then the endomorphisms of the S–local sphere in 𝒮ét(X) are given by étale hypercohomology with coefficients in the S–local classical sphere spectrum:


This confirms a conjecture of Morel.

The primary novelty of our argument is that we use the pro-étale topology of Bhatt and Scholze (Astérisque 369 (2015) 99–201) to construct directly an invertible object 1̂p(1)[1]𝒮(Xét)p with the property that ep(1̂p(1)[1])Σ𝔾m𝒮ét(X)p.


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Tom Bachmann. "Rigidity in étale motivic stable homotopy theory." Algebr. Geom. Topol. 21 (1) 173 - 209, 2021.


Received: 29 May 2019; Revised: 24 March 2020; Accepted: 11 May 2020; Published: 2021
First available in Project Euclid: 16 March 2021

Digital Object Identifier: 10.2140/agt.2021.21.173

Primary: 14F20 , 14F42

Keywords: étale cohomology , motivic homotopy theory

Rights: Copyright © 2021 Mathematical Sciences Publishers


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Vol.21 • No. 1 • 2021
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